The Paradox of Infinity

  • Thread starter RuroumiKenshin
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  • #1
What is infinity? How can something infinite expand? Once and for all, how can we conclude that the universe is infinite? The fact that the universe is infinite may contradict brane theory. Consequently, could it be tenable to suggest that this brane is a subuniverse?
Of course, the idea of an infinite universe could just be based on the inifite boundary theory proposed by Stephen Hawking.
 

Answers and Replies

  • #2
ObsessiveMathsFreak
406
8
I feel it important, nay nessesary, to point out, right before this discussion begins that....

Infinity is NOT a number.

It is NOT a number.

Thank you. Please bear this in mind.
 
  • #3
Mentat
3,918
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Majin, you should perhaps have posted this in the Philosophy Forum.

Infinity is the concept of something that goes on forever.

Something infinite cannot expand (IMO).

We can never conclude that the universe is infinite, beyond question.

No, branes are not "sub-universes". Branes are mathematical additions of dimensions (AFAIK).
 
  • #4
plus
178
1
Without meaning to be insulting, but how can someone not yet able to manipulate equations claim to be able to reason about brane theory?
 
  • #5
drag
Science Advisor
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What's all this talk about infinity being
paradoxical, all the time? What's so paradoxical
about it ? (I guess this should indeed be in the
philosophy forum.)

What's paradoxical about expanding infinity ?

Live long and prosper.
 
  • #6
I knew I should've put this is in the philosophy forum! *slaps head* tsk, tsk.

Anyhow, infinity is paradoxical because, as Mentat pointed out, something infinite is unable to expand. It goes on, and on, forever so how can something infinite expand?
 
  • #7
Originally posted by plus
Without meaning to be insulting, but how can someone not yet able to manipulate equations claim to be able to reason about brane theory?

I'm working on it. Any how, I love doing math, because its just like a logic problem, and I love those too.

plus, what you said was not insulting. The truth is never,AFAIK, insulting.
 
  • #8
drag
Science Advisor
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Greetings !
Originally posted by MajinVegeta
Anyhow, infinity is paradoxical because, as
Mentat pointed out, something infinite is
unable to expand. It goes on, and on, forever
so how can something infinite expand?
Well, if it's only spacial expansion for example
then the distances between all objects will
simply grow.

Do understand me people - I'm not, at the moment,
denying the concept of real infinity as having
its own paradox of some kind, but I have yet
to see any clear fomalization of such a paradox.

Live long and prosper.
 
  • #9
Mentat
3,918
3
Originally posted by plus
Without meaning to be insulting, but how can someone not yet able to manipulate equations claim to be able to reason about brane theory?

What/who are you talking about?
 
  • #10
Mentat
3,918
3
Originally posted by drag
Greetings !

Well, if it's only spacial expansion for example
then the distances between all objects will
simply grow.

No, no, I was talking about the idea that the spacial dimensions themselves are expanding, not just that things are getting farther apart.
 
  • #11
plus
178
1
Something infinite can expand.

Consider the set of the numbers A = {1, 2, 3, ......}, and now consider the set of A and 1/2, so it is B := {1/2, 1, 2, 3, 4, 5, 6, 7, ..}.

Then there is one number in B which is not in A. This could be said to be an expansion of A.

This I believe is a counterexample to one of Mentat's ascertations:

'Something infinite cannot expand',

although he needs say more precisely what he means by this.
 
  • #12
This infinity talk remindes me of cardinality, countable, and uncountable sets. It's a really cool topic in mathematical analysis. In a sense, not all infinities are equal, i.e., not all sets with infinite number of elements can be enumerated. For instance, the interval (0,1) or any finite interval on the real line is larger than the set of all integers.
 
  • #13
drag
Science Advisor
1,100
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Greetings !
Originally posted by Mentat
No, no, I was talking about the idea that
the spacial dimensions themselves are
expanding, not just that things are
getting farther apart.
That IS what I meant. Plus gives an appropriate
example here.

For example, suppose that you draw coordinates
and assign a number to every area of specific
size. Then multiply your infinite set of areas
by 2 and you'll see that the amount of areas
is now twice the previous amount - between say
your house and the nearest store.

It may be conceptually difficult, but if you
consider that space is infinite you don't have
to worry about what's outside at all - simply
don't bother yourself with that thought.
(I must note that I find it difficult, at this time,
to think of a more solid formalization of this idea,
so perhaps there is a fundumental mistake of some
kind in this line of reasoning.)

Live long and prosper.
 
  • #14
Mentat
3,918
3
Alright, let me explain: I am not talking about mathematical infinities. I know that there can be more than one infinity, in maths, and that there can be "bigger" infinities. However, I'm talking about space itself. If you take a space that is infinite (as in, having no end), how could you possibly add more space to this? Even if you could, (though I don't think you could) your result would be the same as before (infinity), and thus you haven't made it any bigger at all.
 
  • #15
drag
Science Advisor
1,100
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Well, I wasn't talking about math, just using
it as an example. The question is - what's the
difference ? Is there any ? And why ?

Live long and prosper.
 
  • #16
If it is already infinite, it can't quite get bigger. Infinity is having no end, eternal. Infinity describes endlessness in the whole so you can't add anything to it.
 
  • #17
dav2008
Gold Member
612
1
ok ok ..right...BUT ..did you see that example?

Take all even numbers...you would agree that there are an infinite amount of them....But you can always add an odd number to that set of inifinite numbers...just like "plus" said...

Its more mathematical in that sense...not physical..
 
  • #18
Yes, I know that in mathematics infinity can be added with another infinity and the sort. BUT, my question is in the physical sense. We can't run away from the physical world.
 
  • #19
Originally posted by MajinVegeta
Yes, I know that in mathematics infinity can be added with another infinity and the sort. BUT, my question is in the physical sense. We can't run away from the physical world.

I believe that what separates the mathematical universe from the physical universe is practicality. Practicality tells us that in order to exist, the universe must have physical laws. These laws (speed of light, for example) impose finite barriers in an attempt to give order to the universe. Even though these laws do prevent many physical infinities (max speed, min temp, etc), the fact that the universe is in constant motion creates many new infinities.

A common infinity is the coastline of great britain. It is impossible to obtain an exact answer for this, for to do so one must measure at increments infinitely less than planck length, which would require so much energy as to warp space enough to disrupt their results. Not to mention the uncertainty principle...

This is not to say that Britain does not have a definite coastline at a specific increment of time. It does, yet practicality (that word again) prevents us from saying so, because WE cannot obtain this measurement. The physical universe has a human element to it that the mathematical universe lacks. Einstein said that "Only two things are infinite: the universe and human stupidity, and I'm not sure about the former." So perhaps the universe is finite, and humans are so infinitely stupid that they cannot comprehend this.

Perhaps this thread belongs in the "philosophy" forum, but oh well.
 
  • #20
Yes, maybe I should ask a moderator to move this thread to philosophy.

Anyway, you bring up an excellent point that has been neglected. Although, it occurs to me that the concept of human stupidity being the obstacle of our understanding of the universe seems be undefined. I came upon this conclusion on the premise that everything is relative. If you were traveling at c, the external world would become blue shifted. But to an external observer, you are the one who is blue shifted. So, you, and the observer are both correct for the whole matter is anisotropic and each of you has the right to your observations. Now, from our perspective, the universe is infinite. Possibly, to an "external" observer, the universe, perhaps, is not infinite. I doubt stupidity has anything to do with the matter.
 
  • #21
if i may say what i think of infinity?

infinity is the only other singular point as zero is in respect to positive and negative.make XOY system.it's flat surface/plane with 4 infinity points (2 on each axis).now try to do this:
shift x and y for infinity.you'll get that they both begin with -0 and end with +0 while there is only one infinity in the center.i think that XOY should transform into sphere with radius inf/2 where zero and inf are diametricaly oposite.
 
  • #22
To make my point defintely established:

THIS THREAD IS MAINLY QUESTIONING THE PHYSICAL PARADOXES OF INFINITY. NOT MATHEMATICAL PROSPECTS.
 
  • #23
drag
Science Advisor
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Originally posted by MajinVegeta
THIS THREAD IS MAINLY
QUESTIONING THE PHYSICAL PARADOXES OF INFINITY.
NOT MATHEMATICAL PROSPECTS.
Indeed. But, should there be a fundumental
difference ? And why ?
(I adressed this above, I believe.)

Live long and prosper.
 
  • #24
I want to expand on the definition of infinity (no pun intended):

In my mind at least, infinity is something that, not only goes on forever, but also has gone on for ever, well as far as the Physical definition goes (refering to the universe)
 
  • #25
plus
178
1
There is no need for such large writing.

Mathematics is used to describe the physical universe. Things which are infinite CAN get larger.

A physical example of something getting larger.

Consider the domain in which you are allowed to move in cylindrical coordinates.
Now suppose that at a time t, you are allowed to move in the range
0< theta < t, where theta is the angle in degrees and that there is no restriction on the vertical range or the horizontal distance.

Then up to t=360, although the domain is infinite, it is still expanding. This is not "just a mathematical infinity" it is a practical one, which I have described the only way possible, using mathematics.


NB I could have easily chosen the function so that it increased for ever, but chose this one as it is easier to see what is happening.

Surely you will see now!!!
 
  • #26
Originally posted by Stormy
I want to expand on the definition of infinity (no pun intended):

In my mind at least, infinity is something that, not only goes on forever, but also has gone on for ever, well as far as the Physical definition goes (refering to the universe)

That contradicts the big bang. Apparently, you implied that infinity has no beginning.

Could it be that infinity describes state of being?
 
  • #27
Originally posted by MajinVegeta
That contradicts the big bang. Apparently, you implied that infinity has no beginning.

Could it be that infinity describes state of being?

Actually (and quite technically speaking), at the instant of the big bang, the universe did occupy infinite volume. If we imagine the universe beginning as a singularity (which is not proven, of course), the big bang marked the instant that the universe began expanding, not when it was a singularity. There is a very tiny difference here, but the only universe that has been measured is the infinite universe (after the first increment of planck time), and the big bang marks this increment, not the time before it.

This is, of course, a small argument, but I still feel Stormy poses an interesting question. After all, we really don't know what the universe was before it was infinite.
 
  • #28
wimms
496
0
big bang, big-bang. Bigbangboo. We have this cute law: energy cannot be created nor destroyed. Thats it - no beginning of universe. Bigbang might be beginning of our cosmology, but not universe.

If you don't accept mathmatics, then this is inherently philosophical question. But I'd like to point out that infinity is NOT a number. You can't apply arithmetics on it, only on a number IN that infinite set. Infinity as unique concept has no meaning and no logical sense by our standards. Its more like direction rather than target.

Infinity is intimately related to concept of Nothing. If universe were finite, it would mean there are boundaries. By definition, beyond these boundaries there must be Nothing. Now ponder for a moment, what is Nothing? Say you have two things separated by Nothing, not space, not stuff, not anything. Is that possibly detectable? Nope. If there is nothing separating two things, they must be touching. If there are 2 universes separated by Nothing, they must be touching, lumping into single Universe. Therefore it is not possible to separate by Nothing.

Now we can say that Universe must be infinite just because Nothing is not logically detectable in principle. And from here, nice reasoning of existence, as logical outcome of impossiblity of state of Nothing.

How can it expand? Can ONE thing be infinite? hehe. it can. Can it split? it can. Can it split forever? why not. Can its multiplicity increase forever? it can. Can it be perceived as expnsion? any time.
 
  • #29
drag
Science Advisor
1,100
1
Greetings !
Originally posted by wimms
big bang, big-bang. Bigbangboo. We have
this cute law: energy cannot be created
nor destroyed. Thats it - no beginning of
universe. Bigbang might be beginning of
our cosmology, but not universe.

If you don't accept mathmatics, then this
is inherently philosophical question.
First of all, it's physics not mathematics.
Second, all the physical laws currently
recognized are limmited by the BB and no
one, so far, attempted to expand them to that
point and beyond because we simply have no idea
about it, again - so far.
Originally posted by wimms
But I'd like to point out that infinity is
NOT a number. You can't apply arithmetics on
it, only on a number IN that infinite set.
Infinity as unique concept has no meaning and
no logical sense by our standards. Its more
like direction rather than target.
I believe a mathematician would disagree
with some of that, but I'm not one so I'll
just keep my mouth shut.
Originally posted by wimms
Infinity is intimately related to concept
of Nothing. If universe were finite, it would
mean there are boundaries.
If the Universe's dimensions were rings
for example - where are the limmits ?
Higher dimensions ?
Originally posted by wimms
By definition, beyond these boundaries there
must be Nothing. Now ponder for a moment,
what is Nothing?
How about - not anything. Makes a difference ?
Originally posted by wimms
Now we can say that Universe must be infinite
just because Nothing is not logically
detectable in principle. And from here,
nice reasoning of existence, as logical
outcome of impossiblity of state of Nothing.
What is your definition of logic ?
Is logic (in your definition :wink:)the absolute
guide of the Universe ? Or maybe, your logic
can still outsmart you (like relativity outsmarted
physicts before - and without logical discrepancies) ?

More to the point - the Universe is not logical. :wink:

Live long and prosper.
 
  • #30
Mentat
3,918
3
Originally posted by drag
Well, I wasn't talking about math, just using
it as an example. The question is - what's the
difference ? Is there any ? And why ?

Live long and prosper.

Well, you were talking about a mathematical concept, the adding of infinities. I was merely showing that this mathematical concept is less present in the "real" (physical) world than a straight line (this is a hyperbole, please let there be no debate on whether one impossibility is more likely than another).

Anyway, did you read my previous post? Is there anything wrong with my reasoning?
 
  • #31
plus
178
1
Originally posted by drag
Greetings !

I believe a mathematician would disagree
with some of that, but I'm not one so I'll
just keep my mouth shut.

Correct.

There is a huge amount of research done into the field of infinity, and I have studied some of it. You may want to find out about the extended real line and some really rigorous work there.

If we are to talk rigorously about infinity, then we should all have Phds in maths. We cannot hope to get to grips with some of the facets of it in this thread, which is written mostly by laypeople.

Rigorous mathematics and physics cannot be handled by conceptions alone. It needs to be accompanied by technical detail.
 
  • #32
Could you apply Zeno's paradox to understanding infinity?
 
  • #33
drag
Science Advisor
1,100
1
Originally posted by Mentat
Alright, let me explain: I am not talking
about mathematical infinities. I know that
there can be more than one infinity, in maths,
and that there can be "bigger" infinities.
However, I'm talking about space itself.
If you take a space that is infinite (as in,
having no end), how could you possibly add
more space to this? Even if you could,
(though I don't think you could) your result
would be the same as before (infinity), and
thus you haven't made it any bigger at all.
Like I said before that message - things inside
will get farther or closer apart. You'll have
the symptoms of growth.

What you seem to be bothered by are the borders,
but infinity has none, does it ?

Live long and prosper.
 
  • #34
You don't need a math PhD to play with infinities rigorously... basic courses in set theory and analysis should do. :wink: You can perform arithmetic on infinities in several senses -- cardinal and ordinal arithmetic, by using systems such as the hyperreals or surreals, and prob some others I don't know about. But back to the original question....

When we talk about the expansion of an infinite universe, we don't mean its "total size" gets bigger in the way, say, a balloon gets bigger as we blow it up. We mean that the distances between objects in the infinite universe gets larger. So if there are originall objects at {..,-2,-1,0,1,2,...} they will move to {...,-4,-2,0,2,4,..} after some time, and continue to get further apart.
 
  • #35
wimms
496
0
Originally posted by drag
First of all, it's physics not mathematics.
Second, all the physical laws currently
recognized are limmited by the BB and no
one, so far, attempted to expand them to that
point and beyond because we simply have no idea
about it, again - so far.
Ah, now I'm lost. Why did you find important to object that so strongly?
I believe a mathematician would disagree
with some of that, but I'm not one so I'll
just keep my mouth shut.

If the Universe's dimensions were rings
for example - where are the limmits ?
Higher dimensions ?
ok, lets leave infinities to Phds in math. They feel touched when laymen talk about their pets. Although extended real line specifically warns that infinity is not a number and should never be treated as number, but only as function.

In regards to rings, what you mean by ring? 2D object? 2D planes are infinite, ring is only subset.

How about - not anything. Makes a difference ?

What is your definition of logic ?
Is logic (in your definition :wink:)the absolute
guide of the Universe ? Or maybe, your logic
can still outsmart you (like relativity outsmarted
physicts before - and without logical discrepancies) ?

More to the point - the Universe is not logical. :wink:
not anything vs nothing - no difference. There is no reason to exchange definitions of logic, that will go too far off topic. There is probably also huge research in that area constantly heating up. Suffice it to say that logic is the basis of any and all proofs you'd ever heard of. My application of logic may be flawed, in which case you may show my error. I reached such thought as I showed, and unless you can show my error, I see no reason to be sarcastic.

Universe not logical? kidding right? Some argue that universe is not only logical, but rather IS the thing itself.
 

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